Summary
Let the random variablesX1,X2, ...,X n be generated by the first-order autoregressive modelX i =θX i−1 +e i wheree i ,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ2, and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators of θ as well as a local asymptotic minimax bound of the risks of estimators.
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References
Akahira, M. (1976). On the asymptotic efficiency of estimators in an autoregressive process,Ann. Inst. Statist. Math.,28, 35–48.
Akahira, M. and Takeuchi, K. (1981). Asymptotic efficiency of statistical estimators: Concepts and higher order asymptotic efficiency,Lecture Notes in Statistics, #7, Springer-Verlag.
Begun, Janet M., Hall, W. J., Huang, Wei-Min and Wellner, Jon A. (1983). Information and asymptotic efficiency in parametric-nonparametric models,Ann. Statist.,11, 432–452.
Beran, R. (1976). Adaptive estimate for autoregressive process,Ann. Inst. Statist. Math.,28, 77–89.
Beran, R. (1977). Robust location estimates,Ann. Statist.,5, 431–444.
Bickel, P. J. (1982). On adaptive estimation,Ann. Statist.,10, 647–671.
Hájek, J. (1970). A characterization of limiting distributions of regular estimates,Zeit. Wahrscheinlichkeitsth.,14, 323–330.
Hájek, J. (1972). Local asymptotic minimax and admissibility in estimation,Proc. Sixth Berkeley Symp. Math. Statist. Prob.,1, 175–194, UC at Berkley.
Huang, Wei-Min (1982). Parameter estimation when there are nuisance functions,Technical Report 82/08, University of Rochester, Rochester, New York.
Ibragimov, I. A. and Hasminskii, R. Z. (1981).Statistical Estimation: Asymptotic Theory, Springer-Verlag, New York.
Inagaki, N. (1970). On the limiting distribution of a sequence of estimators with uniformity property,Ann. Inst. Statist. Math.,22, No. 1, 1–13.
Kabaila, P. (1983). On the asymptotic efficiency of estimators of the parameters of an ARMA process,J. Time Series Analysis,4, 37–47.
LeCam, L. (1969).Theorie Asymptotique de la Decision Statistique, Les Press de l'Universite de Montreal.
Levit, B. Ya. (1975). On the efficiency of a class of non-parametric estimates,Theory Probab. Appl.,20, 723–740.
Millar, P. W. (1979). Asymptotic minimax theorems for the sample distribution function,Zeit. Wahrscheinlichkeitsth.,48, 233–252.
Pfanzagl, J. (1982). Contributions to a general asymptotic statistical theory,Lecture Notes in Statistics,#13, Springer-Verlag.
Wellner, Jon A. (1982). Asymptotic optimality of the product-limit estimator,Ann. Statist.,10, 595–602.
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Huang, WM. A characterization of limiting distributions of estimators in an autoregressive process. Ann Inst Stat Math 38, 137–144 (1986). https://doi.org/10.1007/BF02482506
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DOI: https://doi.org/10.1007/BF02482506